The International Mathematical Olympiad (IMO) is a globally recognized competition that challenges high school students with complex mathematical problems. Among its four categories, geometry stands out as the most consistent in structure, making it more accessible and well-suited for fundamental reasoning research. Automated geometry problem-solving has traditionally followed two primary approaches: algebraic methods, such as Wu’s method, the Area method, and Gröbner bases, and synthetic techniques, including Deduction databases and the Full angle method. The latter aligns more closely with human reasoning and is particularly valuable for broader research applications.
Previous research introduced AlphaGeometry (AG1), a neuro-symbolic system designed to solve IMO geometry problems by integrating a language model with a symbolic reasoning engine. From 2000 to 2024, AG1 achieved a 54% success rate on the issues, marking a significant step in automated problem-solving. However, its performance was hindered by limitations in its domain-specific language, the efficiency of its symbolic engine, and the capability of its initial language model. These constraints prevented AG1 from surpassing its current accuracy despite its promising approach.
AlphaGeometry2 (AG2) is a major advancement over its predecessor, surpassing the problem-solving abilities of an average IMO gold medalist. Researchers from Google DeepMind, the University of Cambridge, Georgia Tech, and Brown University expanded its domain language to handle complex geometric concepts, improving its coverage of IMO problems from 66% to 88%. AG2 integrates a Gemini-based language model, a more efficient symbolic engine, and a novel search algorithm with knowledge sharing. These enhancements boost its solving rate to 84% on IMO geometry problems from 2000-2024. Additionally, AG2 advances toward a fully automated system that interprets problems from natural language.
AG2 expands the AG1 domain language by introducing additional predicates to address limitations in expressing linear equations, movement, and common geometric problems. It enhances coverage from 66% to 88% of IMO geometry problems (2000–2024). AG2 supports new problem types, such as locus problems, and improves diagram formalization by allowing points to be defined using multiple predicates. Automated formalization, aided by foundation models, translates natural language problems into AG syntax. Diagram generation employs a two-stage optimization method for non-constructive problems. AG2 also strengthens its symbolic engine, DDAR, for faster and more efficient deduction closure, enhancing proof search capabilities.
AlphaGeometry2 achieves a high solve rate on IMO geometry problems from 2000–2024, solving 42 out of 50 in the IMO-AG-50 benchmark, surpassing an average gold medalist. It also solves all 30 hardest formalizable IMO shortlist problems. Performance improves rapidly, solving 27 problems after 250 training steps. Ablation studies reveal optimal inference settings. Some issues remain unsolved due to unformalizable conditions or a lack of advanced geometry techniques in DDAR. Experts find its solutions highly creative. Despite limitations, AlphaGeometry2 outperforms AG1 and other systems, demonstrating state-of-the-art capabilities in automated problem-solving.
In conclusion, AlphaGeometry2 significantly improves upon its predecessor by incorporating a more advanced language model, an enhanced symbolic engine, and a novel proof search algorithm. It achieves an 84% solve rate on 2000–2024 IMO geometry problems, surpassing the previous 54%. Studies reveal that language models can generate full proofs without external tools, and different training approaches yield complementary skills. Challenges remain, including limitations in handling inequalities and variable points. Future work will focus on subproblem decomposition, reinforcement learning, and refining auto-formalization for more reliable solutions. Continued improvements aim to create a fully automated system for solving geometry problems efficiently.
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